Probability 2 Dice Question

A pair of dice is rolled once, what is the probability that neither a doublet nor the sum of 10 will appear

2. Relevant equations

P(A) = 1 - P(A')

Demorgans law
(AUB)c = Ac ∩ B c

3. The attempt at a solution

I know how to do the solution through brute force and listing out all the possible scenarios:
(1,1), (1,2), (1,3) ...... (6,6) and doing it like that. But i wanted a more algebraic approach so I tried this:

A pair of dice is rolled once, what is the probability that neither a doublet nor the sum of 10 will appear

2. Relevant equations

P(A) = 1 - P(A')

Demorgans law
(AUB)c = Ac ∩ B c

3. The attempt at a solution

I know how to do the solution through brute force and listing out all the possible scenarios:
(1,1), (1,2), (1,3) ...... (6,6) and doing it like that. But i wanted a more algebraic approach so I tried this:

It does work, and it is essentially what I used:
[tex] \text{desired answer} = \text{P}(A^c \cap B^c) = 1 - \text{P}(A \cup B)[/tex]
If you evaluate ##\text{P}(A \cup B)## correctly you will get the right answer.